This is a tough one. I don't know why the blanket stitch is breaking. Can you adjust the width/length of the stitch to see if that helps? Try a sample first.
You might try a light cut away stabilizer on the back. Then cut around and leave in the towel.
There may be some tips here that help. It is for machine EMB but perhaps a tip or two to find a solution.
https://www.emblibrary.com/EL/ELProj...ry=Stabilizers
Here too...
https://www.emblibrary.com/EL/ELProj...ory=Fabrics101